Lesson 4 solved the telegrapher’s equations, and Lesson 5 read the physics out of the solution. On a lossless line the voltage is the sum of two waves: one with amplitude \(V_0^+\) traveling in the \(+z\) direction, toward the load, and one with amplitude \(V_0^-\) traveling in the \(-z\) direction, back toward the source. Each wave carries a current equal to its voltage divided by the characteristic impedance \(Z_0\). The minus sign on the second current term appears because the backward wave carries its current in the \(-z\) direction. Until today, nothing set the size of \(V_0^-\). The load does.

Lesson 3 introduced the coaxial line, an inner conductor of radius \(a\) inside a shield of radius \(b\), and quoted its per-unit-length inductance and capacitance. Deriving them requires the fields between the conductors: Lesson 25 derives \(C'\) from Gauss’s law, and Module 3 derives \(L'\) from Ampere’s law. Here we use the results. The inductance comes from the magnetic field that circles the inner conductor, and the capacitance comes from the radial electric field between the conductors, so both depend only on the ratio \(b/a\) and on the material between them.

Dividing the two and taking the square root gives the characteristic impedance. The \(\eta\) in that result is the wave impedance of the dielectric, \(\eta = \eta_0/\sqrt{\epsilon_r}\) with \(\eta_0 \approx 377\ \Omega\), and for a nonmagnetic dielectric the expression simplifies to the \(60\ \Omega\) form. Because \(Z_0\) depends on the logarithm of \(b/a\), it changes slowly with the dimensions: doubling \(b/a\) adds only \(41.6\ \Omega/\sqrt{\epsilon_r}\).

Now we connect a load impedance \(Z_L\) at \(z = 0\), the convention from Lesson 3. The load obeys Ohm’s law: the voltage across it divided by the current through it is \(Z_L\). That voltage and current are the line’s voltage and current at \(z = 0\), where every exponential in the wave solution equals one. So the load fixes a relationship between the two wave amplitudes. A single incident wave, with \(V_0^- = 0\), has a voltage-to-current ratio of exactly \(Z_0\), so it can satisfy the load only when \(Z_L = Z_0\). For any other load, a reflected wave must appear to make the ratio come out right.

To solve the boundary condition for the ratio of the amplitudes, we multiply both sides by \(V_0^+ - V_0^-\), collect the \(V_0^+\) terms on the left and the \(V_0^-\) terms on the right, and divide. The result is the reflection coefficient \(\Gamma\), the ratio of the reflected voltage wave to the incident one at the load. It depends only on the load and the line, not on the source or the length of the line. Because \(Z_L\) can be complex, so can \(\Gamma\). We write it as a magnitude and a phase, \(\Gamma = \vert\Gamma\vert e^{j\theta_\Gamma}\), where \(\vert\Gamma\vert\) says how much of the wave comes back and \(\theta_\Gamma\) says how far the reflected wave is shifted in phase. A passive load cannot return more power than it receives, so \(\vert\Gamma\vert\) never exceeds one.

Four loads are worth knowing by heart. A matched load absorbs the whole incident wave, and nothing reflects. A short circuit forces the voltage at the load to zero, so the reflected wave must cancel the incident one exactly: \(\Gamma = -1\). An open circuit forces the current to zero, so the reflected voltage adds to the incident voltage: \(\Gamma = +1\). A pure reactance stores energy but cannot dissipate any, so it returns all of it; the magnitude is one, and only the phase depends on \(X\). Every other load falls somewhere between these cases.

The transmission coefficient \(\tau\) is the ratio of the voltage across the load to the incident voltage. At \(z = 0\) the load voltage is the sum of the two waves, so \(\tau = 1 + \Gamma\). For the \(125\ \Omega\) load, \(\tau = 1.429\): the load voltage is 43% larger than the incident voltage. That is not a power gain. The load current is smaller than the incident current by a larger factor, because the reflected current subtracts, and the power delivered to the load is less than the incident power. On a short circuit, \(\tau = 0\); on an open circuit, \(\tau = 2\), and the voltage at the open end doubles.

With \(\Gamma\) known, one amplitude describes the whole line. Factoring \(e^{-j\beta z}\) out of the sum, whose magnitude is one, leaves the magnitude of the voltage at any point. The term \(\Gamma e^{j2\beta z}\) is the reflection coefficient at position \(z\): the ratio of the reflected wave to the incident wave there, which has the same magnitude as \(\Gamma\) everywhere on a lossless line but rotates in phase by \(2\beta z\), twice the phase delay to the load, because the reflected wave travels to the load and back. Where that term points in the same direction as one, the two waves add; where it points opposite, they subtract.

To find how the magnitude varies, we square it, which turns it into the product of the term and its complex conjugate. Multiplying out, the two cross terms are complex conjugates of each other, and a number plus its conjugate is twice its real part, which gives the cosine. The cosine swings between \(+1\) and \(-1\), so the squared magnitude swings between \(1 + \vert\Gamma\vert^2 + 2\vert\Gamma\vert = (1 + \vert\Gamma\vert)^2\), where the two waves are in phase, and \((1 - \vert\Gamma\vert)^2\), where they are out of phase. Taking square roots gives \(V_\text{max}\) and \(V_\text{min}\). The pattern does not move along the line, which is why it is called a standing wave. The phase \(2\beta z\) changes by \(2\pi\) over half a wavelength, so the maxima repeat every \(\lambda/2\), and each minimum sits halfway between two maxima, a quarter wavelength from each. On a matched line, \(\Gamma = 0\), and the voltage magnitude is the same everywhere. On a short or an open, \(\vert\Gamma\vert = 1\), and the minima fall all the way to zero.

The voltage standing wave ratio is the ratio of the largest voltage on the line to the smallest. It carries the same information as \(\vert\Gamma\vert\) but was historically easier to measure: slide a probe along a slotted line and read the maximum and minimum. It runs from one, for a perfect match, to infinity, for a load that reflects everything. For the \(125\ \Omega\) load,

which equals \(125/50\). For any real load, VSWR is the larger of \(Z_L/Z_0\) and \(Z_0/Z_L\), a quick check worth remembering. VSWR discards the phase of \(\Gamma\), so two different loads can share a VSWR and still behave differently along the line.

The phase of \(\Gamma\) sets where the pattern sits. Measure distance from the load toward the source as \(d = -z\). The waves add in phase where the phase of \(\Gamma e^{-j2\beta d}\) is zero, so the first maximum sits at \(2\beta d_\text{max} = \theta_\Gamma\), with \(\theta_\Gamma\) taken between \(0\) and \(2\pi\). A real load larger than \(Z_0\) has \(\theta_\Gamma = 0\) and a maximum right at the load; a real load smaller than \(Z_0\) has \(\theta_\Gamma = 180^\circ\) and a minimum there instead. For the \(50 + j50\ \Omega\) load the maximum sits \(0.088\lambda\) from the load. Measuring the VSWR and the position of the minimum was the classic way to find an unknown load impedance before network analyzers.

Power goes as voltage squared, so the fraction of the incident power that reflects is \(\vert\Gamma\vert^2\), and the fraction the load absorbs is \(1 - \vert\Gamma\vert^2\). The \(125\ \Omega\) load reflects \(0.429^2 = 18.4\%\) of the incident power and absorbs the other 81.6%. The table gives a sense of scale. A VSWR of 1.5 reflects only 4% of the power, and most equipment specifications treat 2:1 as the edge of acceptable. Beyond that, the load wastes the reflected power, and in a transmitter that power returns to the amplifier, which may not tolerate it. The rule of thumb is that VSWR below 2 is a good match for most purposes.

Today the load set \(\Gamma\), and \(\Gamma\) set the standing wave. The same standing wave means the ratio of voltage to current changes along the line, so a source connected at a distance \(\ell\) from the load does not see \(Z_L\). It sees a transformed impedance that depends on \(\ell\). Lesson 7 derives that input impedance and works the special cases: the shorted line, the open line, and the quarter-wave line. Lessons 8 and 9 turn the transformation into a design tool for matching, and Lessons 10 through 13 do the same work graphically on the Smith chart, which is a plot of \(\Gamma\).

Before Lesson 7, read sections 3.15 and 3.17 through 3.19 of Ellingson, Electromagnetics, Vol. 1.